Majorana modes and transport across junctions of superconductors and normal metals

TL;DR

This paper investigates Majorana modes in one-dimensional p-wave superconductor-normal metal systems, analyzing their effects on sub-gap conductance, the influence of interactions and impurities, and proposing an experimental setup with sign-changing pairing.

Contribution

It provides analytical expressions for Majorana mode energy shifts, studies interaction effects via bosonization, and proposes a new experimental realization with sign-changing pairing.

Findings

Energy shifts of Majorana modes oscillate and decay with superconductor length.

Majorana modes cause conductance peaks at specific energies.

Interactions and impurities significantly influence Majorana modes and conductance.

Abstract

We study Majorana modes and transport in one-dimensional systems with a $p$-wave superconductor (SC) and normal metal leads. For a system with a SC lying between two leads, it is known that there is a Majorana mode at the junction between the SC and each lead. If the $p$-wave pairing $\Delta$ changes sign or if a strong impurity is present at some point inside the SC, two additional Majorana modes appear near that point. We study the effect of all these modes on the sub-gap conductance between the leads and the SC. We derive an analytical expression as a function of $\Delta$ and the length $L$ of the SC for the energy shifts of the Majorana modes at the junctions due to hybridization between them; the shifts oscillate and decay exponentially as $L$ is increased. The energy shifts exactly match the location of the peaks in the conductance. Using bosonization and the renormalization group method, we study the effect of interactions between the electrons on $\Delta$ and the strengths of an impurity inside the SC or the barriers between the SC and the leads; this in turn affects the Majorana modes and the conductance. Finally we propose a novel experimental realization of these systems, in particular of a system where $\Delta$ changes sign at one point inside the SC.